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Aalborg Universitet Distributed Cooperative Control of Nonlinear and Non-identical Multi-agent Systems Bidram, Ali; Lewis, Frank; Davoudi, Ali; Guerrero, Josep M. Published in: Proceedings of the 2013 21st Mediterranean Conference on Control and Automation (MED) DOI (link to publication from Publisher): 10.1109/MED.2013.6608810 Publication date: 2013 Document Version Early version, also known as pre-print Link to publication from Aalborg University Citation for published version (APA): Bidram, A., Lewis, F., Davoudi, A., & Guerrero, J. M. (2013). Distributed Cooperative Control of Nonlinear and Non-identical Multi-agent Systems. In Proceedings of the 2013 21st Mediterranean Conference on Control and Automation (MED). (pp. 770-775). IEEE Press. DOI: 10.1109/MED.2013.6608810 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research. ? You may not further distribute the material or use it for any profit-making activity or commercial gain ? You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us at [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from vbn.aau.dk on: September 18, 2016 This paper is a preprint version of the final paper: A. Bidram , F. L. Lewis, A. Davoudi, and J. M. Guerrero “Distributed Cooperative Control of Nonlinear and Non-identical Multi-agent Systems,” 21st Mediterranean Conference on Control and Automation, 2013. Distributed Cooperative Control of Nonlinear and Non-identical Multi-agent Systems Ali Bidram a, Frank L. Lewis a, Ali Davoudi a, and Josep M. Guerrero b a The University of Texas at Arlington Research Institute and Department of Electrical Engineering, The University of Texas at Arlington, TX, email: [email protected], [email protected], [email protected] b Department of Energy Technology, Aalborg University, 9220 Aalborg East, Denmark , e-mail: [email protected] Abstract— This paper exploits input-output feedback linearization technique to implement distributed cooperative control of multi-agent systems with nonlinear and non-identical dynamics. Feedback linearization transforms the synchronization problem for a nonlinear and heterogeneous multi-agent system to the synchronization problem for an identical linear multiagent system. The controller for each agent is designed to be fully distributed, such that each agent only requires its own information and the information of its neighbors. The proposed control method is exploited to implement the secondary voltage control for electric power microgrids. The effectiveness of the proposed control is verified by simulating a microgrid test system. Index Terms— Distributed cooperative control, Inputoutput feedback linearization, Multi-agent systems, Microgrid. I. INTRODUCTION Multi-agent systems, inspired by the natural phenomena such as swarming in insects and flocking in birds, have received much attention due to their flexibility and computational efficiency. In these phenomena, the coordination and synchronization process necessitates that each agent exchange information with other agents according to some restricted communication protocols [1]-[3].Cooperative control schemes for synchronization of multi-agent systems are mainly categorized in to regulator synchronization problems and tracking synchronization problem. In the regulator synchronization problem, also called leaderless consensus, all agents synchronize to a common value that is not prescribed or controllable. In the tracking synchronization problem all agents synchronize to a leader node that acts as a command generator [4]-[8]. In this paper, the tracking synchronization problem for nonlinear and heterogeneous multi-agent systems is of concern. Cooperative control schemes for multi-agent systems with non-identical and nonlinear dynamics are sparse in the literature [9]-[13].This paper exploits input-output feedback linearization to transform the nonlinear heterogeneous dynamics of the agents to linear dynamics. Using feedback linearization, the synchronization problem for nonlinear and non-identical multi-agent systems is transformed to the synchronization problem for linear and identical multi-agent systems. The Lyapunov technique is adopted to derive fully distributed control protocols for each agent, such that each agent only requires its own information and the information of its agents on the communication graph. Electric power microgrids are small-scale power systems containing distributed generators (DG). The dynamics of DGs are nonlinear and non-identical. Microgrids may get islanded form the main power grid. Once islanded, the DG voltage amplitudes start to deviate. To maintain these voltage amplitudes in stable ranges, the so-called primary control is applied. However, primary control may not return the DG voltage amplitudes to the nominal voltage. This function is provided by the secondary control, which compensates for the voltage and frequency deviations caused by the primary control [14]-[19]. The secondary voltage control of microgrids resembles the tracking synchronization of a multiagent system with nonlinear and non-identical dynamics. The effectiveness of the proposed control scheme, hence, is verified by implementing the secondary voltage control of a typical microgrid. II. P RELIMINARIES OF GRAPH T HEORY The communication network of a multi-agent cooperative system can be modeled by a directed graph (digraph). A digraph is usually expressed as =(VG,EG,AG) with a nonempty finite set of N nodes VG={v1,v2,…,vN}, a set of edges or arcs EG VG×VG, and the associated adjacency matrix AG=[aij] N N . In a microgrid, DGs are considered as the nodes of the communication digraph. The edges of the corresponding digraph of the communication network 1 This paper is a preprint version of the final paper: A. Bidram , F. L. Lewis, A. Davoudi, and J. M. Guerrero “Distributed Cooperative Control of Nonlinear and Non-identical Multi-agent Systems,” 21st Mediterranean Conference on Control and Automation, 2013. denote the communication links. In this paper, the digraph is assumed to be time invariant, i.e., AG is constant. An edge from node j to node i is denoted by (vj,vi), which means that node ireceives information from node j. aij is the weight of edge (vj,vi), and aij>0 if (vj,vi) EG, otherwise aij=0. Node i is called a neighbor of node jif (vi,vj) EG. The set of neighbors of node j is denoted as Nj={i│(vi,vj) EG}. For a digraph, if node i is a neighbor of node j, then node jcan get information from node i, but not necessarily vice versa. The in-degree matrix is defined as D=diag{di} N N with d i jNi aij . The Laplacian matrix is defined as L=D-AG. A direct path from node i to node jis a sequence of edges, expressed as {(vi,vk),(vk,vl),…,(vm,vj)}. A digraph is said to have a spanning tree, if there is a root node with a direct path from that node to every other node in the graph [3]. A direct relationship between the dynamics of the outputs yi (t ) and the control inputs ui (t ) is generated by differentiating the yi (t ) . Differentiating the output of the ith agent yields yi L fi hi Lgi hi ui . (3) An auxiliary control vi is defined as vi L fi hi Lgi hi ui . If Lgi hi is invertible over , the control input ui can be expressed as ui ( Lgi hi )1 ( L fi hi vi ). tinued until ui can be written as a function of vi as follows. is the state vector, ui (t ) input, and yi (t ) that fi ( ) : ni ni L f k h L f ( L f k 1h) matrix b. Lgi L fi r 1hi 0, i . fi (0) 0 . output of all nodes to the output of a leader node y0 (t ) , i.e. one requires yi (t ) y0 (t ), i . The leader node can be viewed as a command generator that generates the desired trajectory x0 f 0 ( x0 ) . (2) y0 h0 ( x0 ) The functions f 0 and h0 are assumed to be of class C . Definition 1. For the smooth function h( x) : smooth vector field f ( x) : n n n and , the Lie derivative is h h being the Jacobian maf ( x ) , with x x n , Assumption 1. There exists an r 1 such that is the output ofi node. It is assumed and f ( x) : f , k 1 , where L f h has x been defined in Definition 1 [20]. th ni field n ( L f k 1h) Lgi L fi l hi 0, In the tracking synchronization problem, it is desired to design distributed control inputs ui (t ) to synchronize the trix [20]. smooth a. The agent state dynamics and state dimensions ni do not need to be the same. The usual assumptions are made to ensure existence of unique solutions. defined as L f h and is the control is locally Lipschitz in n Definition 2. For the smooth vector field h( x) : Consider N nonlinear and heterogeneous systems or agents that are distributed on a communication graph with the node dynamics xi fi ( xi ) gi ( xi )ui , (1) yi hi ( xi ) where xi (t ) (5) If Lgi hi is equal to zero, the differentiation process is con- III. SYNCHRONIZATION AND FEEDBACK LINEARIZATION ni (4) for l r 1, i . According to Definition 2 and Assumption 1a, the rth derivative of yi (t ) can be written as yi( r ) L fi r hi Lgi L fi r 1hi ui . (6) Define the auxiliary control vi as vi L fi r hi Lgi L fi r 1hi ui . (7) If Assumption 1b holds, then the control input ui is implemented as ui ( Lgi L fi r 1hi )1 ( L fi r hi vi ). (8) th Equation(7) results in the r -order linear system yi( r ) vi , i. (9) Equation (9) and the first (r 1) derivatives of yi can be written as yi yi ,1 y y i ,1 i ,2 , i. (10) yi ,r 1 vi or equivalently i 2 A i Bvi , i, (11) This paper is a preprint version of the final paper: A. Bidram , F. L. Lewis, A. Davoudi, and J. M. Guerrero “Distributed Cooperative Control of Nonlinear and Non-identical Multi-agent Systems,” 21st Mediterranean Conference on Control and Automation, 2013. where i [ yi yi,1 yi,r 1 ]T , and 0 0 A 0 0 0 0 . 1 0 rr 1 0 0 0 1 0 0 0 0 0 0 0 0 (12) i be the eigenvalues of L G . The matrix H I N A c( L G) BK , Y0 AY0 By0( r ) , where Y0 [ y0 y0 (14) jNi The pinning gain bi 0 is nonzero only for the nodes that are directly connected to the leader node. From (15), the global error vector for graph is written as where T 1 0 L G I r δ, T T N T 2 , e e1T eT2 T eTN trol vector. 0 0 T vN v2 (17) 0 i and matrices Q QT and R RT be positive definite. Let feedback gain K be chosen as (20) K R 1BT P1 , where P1 is the unique positive definite solution of the control algebraic Riccati equation (21) AT P1 P1A Q P1BR1BT P1 0. Then, all the matrices A ci BK , i c 1 are Hurwitz if , where min mini Re(i ) [21]. 2min Assumption 2. The vector y ( r ) 1N y (0r ) , r is bounded so 0 that y(r ) 0 YMr , with YMr a finite but generally unknown bound. Theorem 1. Let the digraph have a spanning tree and bi 0 for at least one root node. Assume that the zero dynamics of each node i Wi (0, i ), i are asymptotically ( I N B) y ( r ) , 0 (18) where y ( r ) 1N y0( r ) . 0 Definition 3. The have a spanning tree and bi 0 for at least one root node. Let ( A , B ) be stabilizable is the global auxiliary con- can be written as ( I N A) digraph to have a spanning tree and bi 0 for at least one root node. , can be written as ( I N A) ( I N B)v, where v v1 Remark 1. Note that A in(12) has r eigenvalues at s=0 and, hence, is unstable. Therefore, Lemma 1 requires that i 0, i , that is L+G is non-singular. This requires the (16) 0 1N Y0 , G=diag{bi}, and δ is the global disagree- ment vector. with c and K , is Hurwitz if and only if all the matrices A ci BK , i are Hurwitz [4], [21]. Lemma 2. Let the digraph (8)such that i Y0 , i . To solve this problem, the cooperative team objectives are expressed in terms of the local neighborhood tracking error (15) ei aij ( i j ) bi ( i Y0 ). (19) 1r . The synchronization problem is to find a distributed vi in e L G I r B, t t0 t f [10]. The following lemmas are required. Lemma 1. Let ( A , B ) be stabilizable. Let the digraph have a spanning tree and bi 0 for at least one root node. Let Using this input-output feedback linearization, the dynamics of each agent is decomposed into the rth-order dynamical system in (11), and a set of internal dynamics denoted as (Slotine & Li, 1991) i Wi ( i , i ), i . (13) The commensurate reformulated dynamics of the leader node in (2) can be expressed as y0( r 1) ]T i (t0 ) Y0 (t0 ) B [0 0 1]Tr1 , are cooperative UUB with respect to Y0 in (14) if there exists a compact set r so that ( i (t0 ) Y0 (t0 )) there exists a bound B and a time t f ( B,( i (t0 ) Y0 (t0 ))) , both independent of t0 , such that stable. Let the auxiliary control vi in (8)be chosen as vi cKei , (22) where c is the coupling gain, and K 1r is the feedback control matrix. Then i are cooperative UUB with respect to Y0 in (14)and all nodes synchronize to Y0 if K is chosen as in (20) and 1 (23) c , 2min where min mini Re(i ) . 3 This paper is a preprint version of the final paper: A. Bidram , F. L. Lewis, A. Davoudi, and J. M. Guerrero “Distributed Cooperative Control of Nonlinear and Non-identical Multi-agent Systems,” 21st Mediterranean Conference on Control and Automation, 2013. Proof: Consider the Lyapunov function candidate 1 (24) V eT P2e, P2 P2T , P2 0, 2 where e is the global error vector in (16). Then V eT P2e eT P2 ( L G I r ) 0 eT P2 ( L G I r ) ( I N A)δ ( I N B)(v y ( r ) ) . 0 (25) Since the digraph has a spanning tree, and bi 0 for at least one root node, L+G is invertible. Therefore, (25) can be written as V eT P2 ( L G I r )(( I N A )(( L G )1 I r )e ( I N B)(v y ( r ) )). (26) IV. T RACKING S YNCHRONIZATION P ROBLEM IN M ICROGRIDS In this section, the feedback linearization-based tracking synchronization method presented in Section III is used to implement the secondary voltage control of microgrids. Figure 1 shows the block diagram of an inverter-based DG. It contains the primary power source (e.g., photovoltaic panels), the voltage source converter (VSC), and the power, voltage, and current control loops. The control loops set and control the output voltage and frequency of the VSC. Outer volt-age and inner current controller block diagrams are elaborated in [22]. The power controller provides the voltage * * references vodi and voqi for the voltage controller, and the 0 The global auxiliary control v can be written as v c( I N K )e. (27) Placing (27) into (26) and considering the fact that ( A B)(C D) ( AC) ( BD) yields V eT P2 I N A c( L G ) BK e eT P2 (( L G ) B) y ( r ) eT P2 He 0 T e P2 (( L G ) B) y ( r ) . 0 (28) From Lemma 1 and Lemma 2, H is Hurwitz. Given any positive real number , choose the positive definite matrix Fig 1. The block diagram of a DG. P2 , such that the following Lyapunov equation holds, P2 H H P2 I Nr . T (29) Placing (29) in (28) yields V 1 T e ( P2 H HT P2 )e eT P2 (( L G ) B) y ( r ) 0 2 2 e I Nr e e T T According to Assumption 2 2 V e e ( P2 (( L G ) B))YMr , 2 then, V 0 if e (30) P2 (( L G ) B) y ( r ) . 0 2 ( P2 (( L G ) B))YMr . (31) (32) Equation (32) shows that the global error vector e is UUB. Therefore, the global disagreement vector δ is UUB and, hence, i are cooperative UUB with respect to Y0 [9]. If zero dynamics are asymptotically stable, then (9) and (22) are asymptotically stable [20]. This completes the proof. □ Remark 2. If y0 0 , the error bound in (32) is zero and the global error vector e is asymptotically stable. (r) operating frequency i for the VSC. Note that nonlinear dynamics of each DG in a microgrid are formulated on its own d-q (direct-quadratic) reference frame. The reference frame of microgird is considered as the common reference frame, and the dynamics of other DGs are transformed to the common reference frame. The angular frequency of this common reference frame is denoted by com . The nonlinear dynamics of the ith DG, shown in Fig. 1, can be written as xi fi (xi ) k i (xi )Di gi (xi )ui , (33) yi hi (xi ) The term Di is considered as a known disturbance. Detailed expressions for fi (xi ) , gi (xi ) , hi (xi ) , di (xi ) , and k i (xi ) are adopted from the nonlinear model presented in [22]. The primary voltage control is usually implemented as a local controller at each DG using the droop technique. Droop technique prescribes a desired relation between the voltage amplitude and the reactive power. The primary voltage control is (34) vo*,magi Vni nQi Qi , * where vo,magi is the voltage set point provided for the inter4 This paper is a preprint version of the final paper: A. Bidram , F. L. Lewis, A. Davoudi, and J. M. Guerrero “Distributed Cooperative Control of Nonlinear and Non-identical Multi-agent Systems,” 21st Mediterranean Conference on Control and Automation, 2013. Rl1 Ll1 nal voltage control. Qi is the filtered reactive power at the DG’s terminal. nQi is the droop coefficient that is chosen based on the reactive power ratings of DGs. Vni is the primary control reference [19]. Loads The secondary frequency control chooses Vni such that the output voltage amplitude of each DG synchronizes to its nominal value, i.e. vo,magi vref . For secondary voltage 0.23 Ω 318 µH Load 1 PL1 (per phase) QL1 (per phase) Rl2 Ll2 12 kW 12 kVAr 0.35 Ω 1847 µH Rl3 Ll3 Load 2 PL2 (per phase) QL2 (per phase) 0.23 Ω 318 µH 15.3 kW 7.6 kVAr control, the input and output in (33) are ui Vni and yi vo,magi , respectively. Considering the nonlinear dynamics of each DG in (33), the input Vni appears in the second derivate of vo,magi , i.e., r 2 in (9). According to the results of Theorem 1, adopting appropriate values for c in (22) , and P1 and Q in (21), the control protocol in (8) and (22) synchronizes the output voltage amplitude of each DG to its nominal value. Fig 2. Single-line diagram of the microgrid test system. V. S IMULATION RESULTS The effectiveness of the proposed secondary voltage control is verified by simulation on an islanded microgrid. Figure 2 illustrates the single line diagram of the microgrid test system. This microgrid consists of four DGs. The lines between buses are modeled as series RL branches. The specifications of the DGs, lines, and loads are summarized in Table I. In this table, K PV , K IV , K PC , and K IC are the parameters of the voltage and current controllers in Fig. 1. The voltage and current controllers used in the following simulation are adopted from [21]. It is assumed that the DGs communicate with each other through the communication digraph depicted in Fig. 3. The associated adjacency matrix of this digraph is 0 0 0 0 1 0 0 0 . AG (35) 0 1 0 0 1 0 0 0 DGs Lines TABLE I SPECIFICATIONS OF THE MICROGRID TEST SYSTEM DG 1 & 2 (45 kVA rating) DG 3 & 4 (34 kVA rating) mP 9.4×10-5 mP 12.5×10-5 nQ 1.3×10-3 nQ 1.5×10-3 Rc 0.03 Ω Rc 0.03 Ω Lc 0.35 mH Lc 0.35 mH Rf 0.1 Ω Rf 0.1 Ω Lf 1.35 mH Lf 1.35 mH Cf 50 µF Cf 50 µF KPV 0.1 KPV 0.05 KIV 420 KIV 390 KPC 15 KPC 10.5 KIC 20000 KIC 16000 Line 1 Line 2 Line 3 Fig 3. Topology of the communication digraph. Fig. 4. DG output voltage magnitudes: (a) when vref=1 pu, (b) when vref=0.95 pu, and (c) when vref=1.05 pu. DG 1 is the only digraph that is connected to the leader node with g1 1 . Since agent outputs need to track constant references, the leader node has no dynamics, i.e. y0 0 .The coupling gain in (22) is c=4 which satisfies(23). The solution of the algebraic Riccati equation in (21) is used to calculate 5 This paper is a preprint version of the final paper: A. Bidram , F. L. Lewis, A. Davoudi, and J. M. Guerrero “Distributed Cooperative Control of Nonlinear and Non-identical Multi-agent Systems,” 21st Mediterranean Conference on Control and Automation, 2013. the feedback control vector Kin(22). In(21), the algebraic 50000 0 Riccati equation parameters are chosen as Q 1 0 and R=0.01. The resulting feedback control vector is K=[2236 67.6]. It is assumed that the microgrid is islanded from the main grid at t=0, and the secondary control is applied at t=0.6 s. Figures 4a, 4b, and 4c show the simulation results when the reference voltage value is set to 1 pu, 0.95 pu, and 1.05 pu, respectively. As seen in Fig. 4, while the primary control keeps the voltage amplitudes stable, the secondary control establishes all the terminal voltage amplitudes to the pre-specified reference values after 0.1 seconds. [9] [10] [11] [12] [13] [14] [15] VI. CONCLUSION This paper proposes a control method for the tracker synchronization problem of multi-agent systems with nonlinear and heterogeneous dynamics. Input-output feedback linearization is used to transform the nonlinear dynamics of agents to linear dynamics. The distributed control inputs are designed such that the synchronization errors are bounded. The proposed control method is used to implement the secondary voltage control for microgrids. The proposed secondary control is fast and uses a sparse and cheap communication network. The effectiveness of the proposed secondary control is verified by simulating a microgrid test system. [16] [17] [18] [19] [20] [21] ACKNOWLEDGMENTS This work is supported in part by the NSF under Grant Numbers ECCS-1137354 and ECCS-1128050, ARO grant W91NF-05-1-0314, AFOSR grant FA9550-09-1-0278, China NNSF grant 61120106011, and China Education Ministry Project 111 (No.B08015). [22] REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] Q. Hui and W. Haddad, “Distributed nonlinear control algorithms for network consensus,” Automatica, vol. 42, pp. 2375–2381, Sept. 2008. W. Ren and R. W. 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